Discrete-Time Chaotic Systems: Applications in Secure Communications

Abstract

The general design for dead-beat and asymptotic synchronizers for a large class of discretetime chaotic systems is proposed. According to whether the form of the transmitter output (drive signal) is linear, nonlinear or the sum of two, different system structures for synchronization discussions are held. Secure communications is then applied taking into consideration to which state in the transmitter masks the message. Examples of different secure communication schemes are discussed, with a comparison given of the various schemes based on the performance of the receivers ability to recover the message. To accomodate the uncertainty existing in the transmitter parameters, an extended Kalman filter (EKF) algorithm is utilized to estimate both the parameters and states when the message is already embedded. To overcome the problem of high error rates of recovered messages while simultaneously estimating parameters, two alternative methods, namely linear output scheme and indirect scheme, are presented to improve the performance. Numerical simulations for secure communications illustrate a binary signal as the message is recovered and recognizable at the receiver's end.

Secure communications of chaotic systems with robust performance via fuzzy observer-based design

“…Therefore, assume the communication system has a parameter mismatch that is exactly known. Inspired by previous works of modulated chaotic communications [3]- [5], the modulated chaotic transmitter with external disturbances are considered and described by fuzzy rules Transmitter Rule is (13) where the message is modulated into the chaotic system via vector designed later; the transmitted signal is embedded with the message ; is an unknown external disturbance with an upper bound; and denotes the disturbance injection matrix. The final output of the fuzzy chaotic transmitter is inferred as (14) (15) where with .…”

“…In the pioneering works of [1] and [2], transmitting messages masked by chaotic signals or modulated by chaotic systems were studied as a form of secure communications. The idea of chaotic masking [1] is to directly add the message in a noise-like chaotic signal at the transmitter, while chaotic modulation [2]- [5] is by injecting the message into a chaotic system as spread-spectrum transmission. Then a coherent detector and some signal processing is thus employed to recover the message from the received signal at the receiver.…”

Chaos, control, and synchronization in some fractional-order difference equations

“…Chaotic discrete-time systems (maps) have received considerable attention over the last two decades due to their many applications in secure communications [1][2][3][4] and control [5]. Numerous maps have been proposed throughout the years including Hénon map [6], Lozi system [7], generalized Hénon map [8], Baier-Klein system [9], Stefanski map [10], Rössler map [11], and Wang map [12].…”

Unknown Input Observers for Message-Embedded Chaos Synchronization of Discrete-Time Systems

“…A sufficient condition for the exponential convergence of the state-error dynamics in the case of Lipschitz nonlinearities are given in [Liao & Huang, 1999]. For discrete-time systems, dead-beat observers can be designed in order to make the matrix involved in the synchronization error equation to be nilpotent [De Angeli, 1995;Lian et al, 2000]. Despite the fact that it depends on the information, the error equation can reach zero after a finite number of steps.…”